I want to show the fly out links (children page links) on the main page. How do I do this?

I am assuming you don't just want them in the left-side links.

Click on the FLY-OUT menu item you want to include a link of on your home page. When the page shows up, copy the website address from the browser's URL field.

Edit the home page (Edit Site Pages, Edit HOME PAGE, move to where you want the link to appear, type "CLICK THIS LINK PLEASE" or whatever you want the link to say.
Select the "Click ..... Please..." text
Click the INSERT LINK button (3rd from right side, 2nd row of icons), Paste the URL you copied from the browser's URL field.
If you want the link to open in a new tab or window, click the TARGET tab, then choose the option you want.
Save the link
Save the home page contents
Test the link.

Thanks! Your instructions were followed and worked perfectly!

Glad I could help!